# STAT 443 Lab 4: SARIMA Processes solution

\$30.00

Original Work ?

## Description

5/5 - (1 vote)

• The lab must be completed in R Markdown. Create a pdf file (via ’knit to pdf’) and use

We have seen both autoregressive (AR) and moving average (MA) models, as well as hybrids
of these known as autoregressive moving average (ARMA) models. While these are quite versatile
models, recall that they are used to model series that are stationary.

1. Without using any mathematical notation, describe in words what it means for a time
series to be stationary.

2. Consider a realization of a process with values given in file lab4data.csv. Read in the
data, coerce it into a ts object, plot and comment on whether the series appears to satisfy
the requirements of stationarity.

3. One common way of removing a trend is to difference the data, where instead of looking
at the time series {xt} we look at {yt} with
yt = ∇xt = xt − xt−1.

(Note this series will have one less term than the original.) Using the data above, determine
and plot the differenced time series {yt}. Comment on the resulting plot and the acf of {yt}.

A useful function here is diff(x, lag=1, difference=1), which returns suitably lagged
and iterated differences. Use R help to learn about options lag and differences.

4. In order to remove a seasonal effect, we could difference over the seasonal period. For
instance, take our de-trended data {yt} and difference again but at a lag equal to the
seasonal period, s. The new series will be
∇syt = yt − yt−s,
where s is the period of the seasonal effect. Note the series will again become shorter,
this time with s fewer terms. Choosing the appropriate value of s here, apply seasonal
differencing to the series from part 3 and plot the resulting series. Plot the acf of ∇syt
.
Does your ∇syt resemble white noise?

5. Suggest which type of model from the SARIMA family you would use for the original data.

6. (Theoretical exercise) We have seen that removing trends and seasonality can be as
simple as differencing successively at different lags. In such cases, suitable models are
integrated ARMA (ARIMA) models, or SARIMA if we include seasonal differencing. One
difficult aspect of these models is describing them in mathematical terms. Let us try
re-using the notation we used with ARMA models to describe these processes.

(a) Firstly, we took our original time series Xt and differenced it to obtain Yt = Xt−Xt−1,
then differenced again at a longer lag to obtain Wt = Yt − Yt−s. Combine these two
operations to express our final model Wt
in terms of the original series Xt
.

(b) Recall the differencing operator B which takes a series and returns the series at lag 1.
For instance, BXt = Xt−1. Express the differenced series Yt
in terms of B and Xt
.

(c) Now if BsYt = Yt−s, express your answer from (a) in terms of B. That is, write Wt
in
terms of B, s, and Xt
.